Can we stop training as soon as epsilon is small?
How much the $Q$-values change does not depend on the value of $\epsilon$, rather the value of $\epsilon$ dictates how likely you are to take a random action and thus take an action that could give rise to a large TD error -- that is a large difference between the returns you expected from taking this action as to what you actually observed. How much the $Q$-value changes depends on the magnitude of this TD error.

$Q$-learning is not guaranteed to converge if there is no exploration. Part of the convergence criteria assumes that each state-action pair will be visited infinitely often in an infinite number of episodes, and so if there is no exploration then this will not happen.